Summary: Multilinear models are models in which the expectation of a multiway array is the sum of products of parameters, where each parameter is associated with only one of the ways. In spectroscopy, multilinear models permit mathematical decompositions of data sets when chemical decomposition of specimens is difficult or impossible. This paper presents a unified description of the models in an array notation. The spectroscopic context shows how to interpret one initialization of the nonlinear least-squares fits of these models. Several examples show that these models can be applied successfully.